I have glass tiles but I guess mine are different. But I will say this if you get 10 glass door companies out there you will get 10 opinions on what needs to happen. The people that don't bring levels and such get rid of them first. They wanted to put support rods across the top of mine and drill into my floor. Here is how mine is constructed maybe you can tweak it to fit your needs. I have a 28" wide x 80" tall 3/8" thick door that has 45 deg hinges that swings off of a 1/2" thick piece of glass that goes from floor to ceiling. There are 2 clips on the wall and one at the top where this glass goes to the ceiling. This holds the panel steady. It is just siliconed at the wall and bottom, no drilling into floor. The other fixed panel is 28" wide x 80" tall 3/8" thick and has only one small attachment at the top corner the rest is siliconed in. It works beautifully and you see the true beauty of the glass not a bunch of hardware and head rails. My Glass tiles are bullet proof as it is glass laminated onto porcelain and it is really tough to drill through but I don't think they should have a problem using the correct drill bit and possibly keeping it wet while drilling with acetone or water to keep it cool. This company might have had problems before and they are now nervous with the situation. I suggest that you keep getting quotes. It's a pain and slows the process but in the end positive results.

Yes in 98% of shower doors the glass can be changed. Probable be about 3/4 the price of a new unit however.

Follow up (I'm OP): GC contacted the lead guy (owner?) of glass co., and within days they were at my house to discuss. They said the door's vertical sweep (which closes to the fixed glass 1/2 panel - see pic) could be replaced with a thinner one. And some glass cos. don't even put a sweep there. They said to totally disregard everything the receptionist said - she had no business diagnosing the problem. The thinner sweep would only be the top half of the door. But we're thinking your eye won't notice that the bottom half has a thicker sweep because there's a 1/2 wall, tiled, there. Get this: The day they showed up the door worked again (pivoted both in and out of the shower)! It was colder - wildly guessing the walls are expanding in heat, or maybe its just a stiff sweep problem (they didn't opine as to cause). Glass co. owner gave me his text and direct dial/vm, and said he'll install a thinner sweep whenever - i.e., if I want, I can wait until its hot again. The only thing I find annoying is that, well, I paid about $3,900 with tax for the glass (HCOL area), and seems I should have a door that fits with the usual sweep. And I'm concerned that if we go with the thinner sweep, maybe the walls will contract when it cold again. But I guess I would rather be done with this and, if it does leak in the winter, deal with that problem then. My 1 year warranty is up Dec. 2017.

Just do a simple sliding door. You choose where and which handles or pulls you want. Even if you only use one door opening the other to start the shower or even cleaning is worth it. You will probably pay more and have less versatility. Frameless doors offer less water containment, Make sure you know what your getting into as they look great in pics until you realize on the sides you have stick on clear plastic strips that yellow and fall off. Get a slim framed door and be done with it. They are repairable and you will have no water issues. Simple clean looking shower don't get the curtain remark.